Traces on General Sets in $\mathbb{R}^{n}$ for Functions with no Differentiability Requirements^††thanks: The author’s research is supported by the NSF award DMS-1716790

Suppose that $\Omega\subseteq\mathbb{R}^{n}$ is an open, not necessarily bounded, set and that $u\in\mathscr{C}(\Omega)$ is uniformly continuous. Though the boundary $\partial\Omega$ is not in the domain of $u$, there is a natural choice for a trace function $Tu\in\mathscr{C}(\partial\Omega)$ that can be identified as the values of $u$ on $\partial\Omega$, since $u$ has a continuous extension to $\overline{\Omega}$. In the context of Sobolev spaces, where $u$ is not necessarily continuous in $\Omega$, the well-known Gagliardo’s theorem [19] states that, with a sufficiently regular boundary, this trace operator can be extended from the space of functions with uniformly continuous derivatives to a continuous linear operator $T:W^{1,p}(\Omega)\to W^{1-\frac{1}{p},p}(\partial\Omega)$, for each $1<p<\infty$. Here $0<\beta<1$ and $W^{\beta,p}(\partial\Omega)$ is the standard Sobolev–Slobodeckij space (Definition 1.3). Generalizations to Sobolev-Slobodeckij and Besov spaces, traces on general closed subsets of $\mathbb{R}^{n}$, Sobolev and Besov spaces on metric spaces, Sobolev spaces with variable exponent, etc. have since been produced (see [14, 31, 33, 34, 39]). This paper is concerned with establishing analogous trace results compatible with a nonlocal framework, where only $p$-integrability is assumed away from the boundary.

There has been a surge of interest in developing and analyzing models that employ a nonlocal operator. Such models can incorporate long-range interactions and multiple scales and can expand the space of admissible solutions to permit singular and discontinuous attributes. Nonlocal models have been successfully employed in a wide variety of contexts, including image processing [24, 32], population and flocking models [8, 40], diffusion [3], phase transitions [5, 23], and material deformation with failure in peridynamics models [41, 42]. For a comprehensive introduction to nonlocal modeling and their analysis, we refer to the monograph [15].

A common class of nonlocal operators have a convolution, or cross-correlation, like structure. At each point $\vectorsym{x}\in\Omega$, the operator uses an integral and an integrable kernel to accumulate weighted data for a function over a neighborhood of $\vectorsym{x}$. This makes them generally insensitive to function values on sets of zero measure and, in particular, to function irregularities across sets of dimension less than $n$. With an appropriate kernel, however, the operator can approximate a differential operator and provide information about the rate of change of a function.

The current work contributes to the rigorous development of a framework in which convolution-like operators, with integrable kernels, can incorporate data on sets with positive codimension. In addition to mathematical interest, there are two primary motivations for these efforts. Both are related to the fact that at points where the support of the kernel extends outside of $\Omega$, the evaluation of the operator requires function values outside of $\Omega$. Thus, for associated nonlocal problems, the analogue of a Dirichlet-type boundary condition is a volume-constraint, where the value of a solution is prescribed on a region of positive measure in $\mathbb{R}^{n}\setminus\Omega$ [16, 38]. It can be a nontrivial issue to identify appropriate volume-constraints when data on only a lower-dimensional set $\Gamma\subseteq\partial\Omega$ is readily available. If the integral operator is responsive to function behavior on $\Gamma$, then one can instead formulate nonlocal problems subject to classical Dirichlet-type boundary conditions. This also facilitates seamless transitions between nonlocal and local system descriptions. There is substantially more computational expense to numerically solve nonlocal equations when compared to their local counterpart. With a nonlocal operator that “localizes” as $\Gamma$ is approached, one can couple a computationally efficient local model to a nonlocal model that confined to a region where it is essential to employ nonlocal operators [12, 43]. The set $\Gamma$ is a lower-dimensional interface between these two regions. In both contexts, it is essential to have a trace theory to ensure well-posedness and mathematical consistency.

In [44], Du and Tian produced some of the first trace results for functions in the nonlocal setting, with operators that have an integrable kernel that concentrates near the boundary. They considered the space $\mathscr{S}(\Omega)\subseteq L^{2}(\Omega)$ consisting of functions satisfying $\|u\|^{2}_{\mathscr{S}(\Omega)}:=\|u\|^{2}_{L^{2}(\Omega)}+|u|^{2}_{\mathscr{S}(\Omega)}<\infty$, where $\Omega\subseteq\mathbb{R}^{n}$ is an open Lipschitz domain and

Here

where $\mathscr{B}(\mathbb{R}^{n})$ denotes the family of Borel subsets of $\mathbb{R}^{n}$, $|E|$ is the Lebesgue measure of $E$, and $\mathcal{B}_{\rho}(\vectorsym{x})$ is the open ball centered at $\vectorsym{x}$ with radius $\rho>0$. The kernel, in the semi-norm $|\cdot|_{\mathscr{S}(\Omega)}$, can be identified as $\vectorsym{x}\mapsto d_{\partial\Omega}(\vectorsym{x})^{-n-2}|\mathcal{B}_{\frac{1}{2}}|^{-1}\chi_{\Psi(\vectorsym{x})}$. Since $d_{\partial\Omega}$ is uniformly positive on any open $\Omega^{\prime}$ compactly contained in $\Omega$, one can only expect $u\in L^{2}(\Omega^{\prime})$. Thus, the space $\mathscr{S}(\Omega)$ includes functions that have no regularity, beyond square-integrability, away from the $\partial\Omega$. As the $\partial\Omega$ is approached, however, the kernel concentrates sufficiently strongly that the behavior of $u$ at $\partial\Omega$ contributes to $|\cdot|_{\mathscr{S}(\Omega)}$. In fact, the singularity is strong enough that $|\cdot|_{\mathscr{S}(\Omega)}$ is sensitive to oscillations in $u$ at the boundary and $\|u\|_{\mathscr{S}(\Omega)}<\infty$ implies that there is a well-defined trace $Tu\in L^{2}(\partial\Omega)$ and, moreover, that $Tu\in W^{\frac{1}{2},2}(\partial\Omega)$. These results have recently been generalized to exponents $1\leq p<\infty$ in [17].

In this paper we vastly expand the class of functions and domains, for which a well-defined trace can be identified for a given $\Gamma\in\mathscr{B}(\partial\Omega)$. Let $1\leq p<\infty$ and $s\in L^{\infty}(\Omega)$ be given. For each $u\in L^{1}(\Omega)$, define

The focus of this paper is on developing a trace theory for functions belonging to the space

A straightforward argument shows that $|\cdot|_{\mathscr{N}^{s(\cdot),p}(\Omega)}:\mathscr{N}^{s(\cdot),p}(\Omega)\to[0,\infty)$, given by $|u|^{p}_{\mathscr{N}^{s(\cdot),p}(\Omega)}:=\nu^{s(\cdot),p}(\Omega;u)$, provides a semi-norm. We easily see that if for some $0<\beta^{\prime}<1$, we have $s\geq\beta^{\prime}$ throughout $\Omega$, then $W^{\beta^{\prime},p}(\Omega)\subseteq B^{1,p}_{\beta^{\prime}}(\Omega)\subseteq\mathscr{N}^{s(\cdot),p}(\Omega)$, with $B^{1,p}_{\beta^{\prime}}(\Omega)$ is a Besov space, as defined in [31]. Moreover, the spaces introduced in [17, 44] are subspaces of the corresponding space $\mathscr{N}^{s(\cdot),p}$, with $s$ constant. (For example, $\mathscr{S}\subseteq\mathscr{N}^{1,2}$.) In fact, Example 1.2 shows that, in general, the containment is strict. Thus, many results currently available in the literature can be viewed as corollaries of the trace theorems established in this paper.

With $\Gamma\in\mathscr{B}(\partial\Omega)$, the main results of the paper provide assumptions on $\Omega$, $\Gamma$, and $s$ (collected in Section 1.3) that ensure the existence of a continuous linear trace operator $T:\mathscr{N}^{s(\cdot),p}(\Omega)\to W^{\beta,p}(\Gamma)$, for some $0<\beta<1$. The primary assumptions on $\Omega$ and $\Gamma$ are (H1) a corkscrew-type condition, (H2) a uniform-connectedness condition, (H3) a constraint on the oscillations of $u$ near $\Gamma$, and (H4) Ahlfors-regularity of $\Gamma$. Loosely speaking, the oscillation constraint is satisfied if $s(\vectorsym{x})\geq a(\overline{\vectorsym{x}})+\beta+\frac{n-\dim(\Gamma)}{p}$, for $\vectorsym{x}\in\Omega$ near $\overline{\vectorsym{x}}\in\Gamma$. The values of $a(\overline{\vectorsym{x}})>0$ depend, in an explicit way, on the approachability of $\overline{\vectorsym{x}}$ from $\Omega$. Following some definitions recorded in Section 1.2, precise statements of the main assumptions and theorems are given in Sections 1.3 and 1.4.

The distinguishing features of this work are the following:

• •

  We provide trace results allowing “very rough” boundaries. In [17] and [44], the set $\Gamma$ is required to be Lipschitz, so that an atlas of Lipschitz transforms are available to “flatten out” $\Gamma$. They connect the rate of change of a function in directions parallel to the flattened boundary to the rate of change in the normal direction. In this paper, a different approach is presented, based on a continuous extension of a function (3) related to the mean values of $u$. The corkscrew and connectedness properties ensure there is a region of approach to points in $\Gamma$ that allows us to work directly with $\Gamma$, without any transformations, and allow $\Gamma$ to have minimal regularity.

  In fact, identifying a natural candidate for a measurable trace does not require any assumptions on $\Omega$ and $\Gamma$ beyond (H1) and (H2). Assuming upper Ahlfors-regularity, we prove $Tu\in W^{\beta,p}(\Gamma)$. If, additionally, there is a lower Ahlfors-regular neighborhood in $\Gamma$, then we can establish the Lesbesgue point property for $Tu$ within that neighborhood; i.e., as $\rho\to 0^{+}$, the mean values of $u$ over $\Omega\cap\mathcal{B}_{\rho}(\overline{\vectorsym{x}})$ converge in the $p$-norm to $Tu(\overline{\vectorsym{x}})$, for a.e. $\overline{\vectorsym{x}}\in\Gamma$. Thus, with respect to the surface measure, $Tu$ agrees a.e. on $\Gamma$ with the strictly defined function associated with $u$.

• •

  Within $\Omega$, the functions need only be $p$-integrable. Given a function $u\in\mathscr{N}^{s(\cdot),p}(\Omega)$ and an open set $\Omega^{\prime}$ compactly contained in $\Omega$, Jensen’s inequality implies

  $$\int_{\Omega^{\prime}}\!|u(\vectorsym{x})|^{p}\mathrm{d}\vectorsym{x}\leq c\int_{\Omega^{\prime}}\!\left(\fint_{\Psi(\vectorsym{x})}\!\!|u(\vectorsym{y})-u(\vectorsym{x})|\mathrm{d}\vectorsym{y}\right)^{p}\mathrm{d}\vectorsym{x}\\ +c\int_{\Omega^{\prime}}\!\left(\fint_{\Psi(\vectorsym{x})}\!\!|u(\vectorsym{y})|\mathrm{d}\vectorsym{y}\right)^{p}\mathrm{d}\vectorsym{x}<\infty,$$

  since $u\in L^{1}(\Omega)$ and $d_{\partial\Omega}(\vectorsym{x})$ is uniformly positive in $\Omega^{\prime}$. Hence, $\mathscr{N}^{s(\cdot),p}(\Omega)\subseteq L^{p}_{\operatorname{loc}}(\Omega)$. As with the space $\mathscr{S}(\Omega)$, however, no additional regularity can be expected for $u$ in $\Omega^{\prime}$, regardless of the behavior of $s$ in $\Omega^{\prime}$. This makes $\mathscr{N}^{s(\cdot),p}$ a viable solution space for nonlocal systems where even discontinuous functions are admissible. For models of phenomena exhibiting sharp transitions or jumps, it is critical to include irregular functions as solution candidates. Moreover, functions in this space possess well-defined fine properties as we approach $\Gamma$.

• •

  The traces are captured on possibly “very thin or porous” sets. The set $\Gamma$ can have non-integer Hausdorff dimension, any positive codimension, and may also possess cusp-like features. To demonstrate how irregular the domain can be, we produce an $\Omega\subseteq\mathbb{R}^{2}$, with a nonrectifiable Ahlfors-regular self-similar set $\Gamma\in\mathscr{B}(\partial\Omega)$ that has Hausdorff dimension $1<t<2$ and has the following property: for every $\overline{\vectorsym{x}}\in\Gamma$ and $\rho>0$, there exists a $\overline{\vectorsym{y}}\in\Gamma\cap\mathcal{B}_{\rho}(\overline{\vectorsym{x}})$ that is only tangentially accessible. In other words, the cone of directions in the approach region for $\overline{\vectorsym{y}}$ degenerates as it is approached. Nevertheless, provided the oscillation constraint is satisfied, there is a trace $Tu\in W^{\beta,p}(\Gamma)$ possessing the Lebesgue point property (see Example 1.1(b), Remark 4(b), and Section 5). We mention the recent interest in developing a theory for elliptic problems with higher-codimensional boundaries [9, 10, 21, 37] and trace theorems and boundary value problems on fractal sets [1, 2, 7].

A more detailed presentation of the main theorems requires some additional definitions. For each $\rho>0$, $E\subseteq\mathbb{R}^{n}$, and $\vectorsym{x}\in\mathbb{R}^{n}$, define $E_{\rho}(\vectorsym{x}):=E\cap\mathcal{B}_{\rho}(\vectorsym{x})$. Given $\tau\geq 0$ and $E\in\mathscr{B}(\mathbb{R}^{n})$, we use $\mathscr{H}^{\tau}(E)$ to denote the $\tau$-dimensional Hausdorff measure of $E$. We use $L(E)$ for the space of Borel-measurable functions. In general, by measurable, we mean Borel-measurable. We now introduce the two main geometric properties needed throughout the paper. A discussion of these definitions with accompanying figures is provided in the next section, following the main assumptions.

In the definition above, without loss of generality, we assume $\varepsilon_{\lambda}\leq\varepsilon_{\lambda^{\prime}}$, if $\lambda\leq\lambda^{\prime}$. Next we recall the definition of the Sobolev-Slobodeckij spaces.

If $E\subseteq\mathbb{R}^{n}$ is a closed set and $0<\beta<1$, then our definition of $W^{\beta,p}(E)$ corresponds to the definition of the Besov space $B^{p,p}_{\beta}(E)$ given in [30] (see also [31]). We note that, in general, Definition 1.3 does not provide the standard Sobolev or Sobolev-Slobodeckij spaces when $\beta\geq 1$. In fact, if $E$ is an open connected set, $\beta\geq 1$, and $u\in W^{\beta,p}(E)$, as defined above, then $u$ is a constant function [6, 11]. It appears to be unknown whether this is also true for $E$ not open.

Finally, we need a bit more notation. For each $0<\lambda<1\leq\theta$ and $\overline{\vectorsym{x}}\in\partial\Omega$, set

(See Fig. 2 below.) Given $\delta>0$, we also define $Q^{\theta}_{\lambda,\delta}(\overline{\vectorsym{x}}):=Q^{\theta}_{\lambda}\cap\mathcal{B}_{\delta}(\overline{\vectorsym{x}})$. For convenience, we define $\alpha\in\mathscr{C}(\mathbb{R}^{2})$ by

We note that $\alpha$ is piecewise bilinear and that $\alpha(\cdot,\theta)$ is increasing, for each $\theta\geq 1$. For the remainder of the paper, we fix the parameters $0\leq t<n$ and $0<\delta_{\Gamma}<1$ and the uniformly bounded measurable functions $s:\Omega\to[0,\infty)$ and $\theta_{\Gamma}:\Gamma\to[1,\infty)$. Put $\overline{\theta}_{\Gamma}:=\sup_{\overline{\vectorsym{x}}\in\Gamma}\theta_{\Gamma}(\overline{\vectorsym{x}})$. For each $0<\delta\leq\delta_{\Gamma}$, define $\underline{s}_{\delta},\underline{\alpha}_{\delta},\underline{\alpha}_{0}:\Gamma\to\mathbb{R}$ by

We observe that the above functions are each measurable and uniformly bounded.

We now list our primary assumptions for $\Omega$, $\Gamma$ and the space $\mathscr{N}^{s(\cdot),p}(\Omega)$. We assume that $\Gamma\in\mathscr{B}(\partial\Omega)$ has $\mathscr{H}^{t}$-dimension. When convenient, we will just write $Q_{\lambda}(\overline{\vectorsym{x}})$ and $Q_{\lambda,\delta}(\overline{\vectorsym{x}})$ for $Q^{\theta_{\Gamma}(\overline{\vectorsym{x}})}_{\lambda}(\overline{\vectorsym{x}})$ and $Q^{\theta_{\Gamma}(\overline{\vectorsym{x}})}_{\lambda,\delta}(\overline{\vectorsym{x}})$, respectively.

• (H1)

  Uniform $\theta_{\Gamma}$-Corkscrew Condition: There exists $0<\lambda_{0},\eta_{0}<1$ such that, for each $\overline{\vectorsym{x}}\in\Gamma$, the set $Q_{\lambda_{0}}(\overline{\vectorsym{x}})$ is an $(\eta_{0},\theta_{\Gamma}(\overline{\vectorsym{x}}))$-corkscrew region for $\overline{\vectorsym{x}}$ with radius $\delta_{\Gamma}$.

• (H2)

  Uniform $\theta_{\Gamma}$-Connectedness Condition: There exists $C_{\Gamma}\geq 1$ such that, for each $\overline{\vectorsym{x}}\in\Gamma$, the set $\Omega_{\delta_{\Gamma}}(\overline{\vectorsym{x}})$ is $(C_{\Gamma},\theta_{\Gamma}(\overline{\vectorsym{x}}))$-connected to $\overline{\vectorsym{x}}$.

• (H3)

  Oscillation Constraints:

  • (H3^′)

    Pointwise Oscillation Constraint at $\Gamma$:

    $$\underline{\alpha}_{0}(\overline{\vectorsym{x}})>-t,\quad\text{ for $\mathscr{H}^{t}$-a.e. }\overline{\vectorsym{x}}\in\Gamma.$$

  • (H3^′′)

    Uniform Oscillation Constraint near $\Gamma$:

    $$\underline{\alpha}_{\Gamma}:=\inf_{\overline{\vectorsym{x}}\in\Gamma}\underline{\alpha}_{\delta_{\Gamma}}(\overline{\vectorsym{x}})>-t.$$

• (H4)

  Ahlfors-Regularity: There exists $A_{\Gamma}\geq 1$ such that, for each $\overline{\vectorsym{x}}\in\Gamma$,

  • (H4^′)

    Upper Ahlfors-Regularity:

    $$\mathscr{H}^{t}(\Gamma\cap\mathcal{B}_{\rho}(\overline{\vectorsym{x}}))\leq A_{\Gamma}\rho^{t},\quad\text{ for all }\rho>0.$$

  • (H4^′′)

    Lower Ahlfors-Regularity:

    $$A^{-1}_{\Gamma}\rho^{t}\leq\mathscr{H}^{t}(\Gamma\cap\mathcal{B}_{\rho}(\overline{\vectorsym{x}})),\quad\text{ for all }0<\rho\leq\operatorname{diam}(\Gamma).$$

We next put the corkscrew and connectedness assumptions into a broader context.

Assumptions (H3) can, in some sense, be interpreted as a requirement that the oscillations in $u\in\mathscr{N}^{s(\cdot),p}(\Omega)$ decay as $\Gamma$ is approached. If $\nu^{s(\cdot,p}(\Omega;u)<\infty$, then there exists $f\in L^{p}(\Omega)$ such that

The left-hand side “measures” the deviation of the values of $u$ from $u(\vectorsym{x})$ over the ball $\Psi(\vectorsym{x})$. Since $t<n$, (H3) implies that, for each $\overline{\vectorsym{x}}\in\Gamma$, there exists a $\delta_{0}>0$ such that $s$ is uniformly positive within the corkscrew region $Q_{\lambda_{0},\delta_{0}}(\overline{\vectorsym{x}})$. Thus, (H3) ensures the oscillations of $u$ dampen in $Q_{\lambda_{0},\delta}(\overline{\vectorsym{x}})$, as $\delta\to 0^{+}$, which allows a well-defined value for $u(\overline{\vectorsym{x}})$ to be identified.

While Jensen’s inequality implies, for all $1\leq q<\infty$, we have

As just discussed, assumption (H3) is a type of decay requirement for the oscillations of $u\in\mathscr{N}^{s(\cdot),p}(\Omega)$ but only in the sense of averages. This still allows the oscillations with uniformly positive amplitude to concentrate on sets of decreasing measure, and its possible for $\nu^{s(\cdot),(p,q)}(\Omega;u)=+\infty$, for every $q>1$.